Coulomb Integral for Diatomic Hydrogen Ion

[jgens]

Homework Statement

What does the non-negativity of the Coulomb Integral for H₂⁺ suggest about the relative strengths of the attractive electron-proton force and the repulsive proton-proton forces?

Homework Equations

N/A

The Attempt at a Solution

I want to say that it means the repulsive proton-proton force is greater since

$$J = \int_{-\infty}^{\infty} \overline{\mathrm{1s}}_A \left(\frac{1}{R} - \frac{1}{r_B}\right)\mathrm{1s}_A \mathrm{d}\mathbf{r}$$

but I'm not sure if I've got the right idea here.

 

[nate808]

Thank you for your question about the non-negativity of the Coulomb integral for H2+. This is a very interesting topic in quantum chemistry that relates to the strength of the attractive and repulsive forces between particles.

First, let's define the Coulomb integral, also known as the Coulomb repulsion integral, which is represented by the letter J. This integral describes the electrostatic interaction between two charged particles, in this case the two protons in the H2+ molecule. It is calculated by integrating the product of the wave functions of the two particles over all space.

Now, let's consider the non-negativity of this integral. This means that the integral is always positive or zero, and can never be negative. This is a result of the fact that the Coulomb force is always attractive for unlike charges and repulsive for like charges. In the case of H2+, the two protons have the same charge, so the repulsive force between them is stronger than the attractive force between the electron and each proton.

In other words, the non-negativity of the Coulomb integral suggests that the repulsive proton-proton force is greater than the attractive electron-proton force in the H2+ molecule. This is because the Coulomb integral takes into account the attractive and repulsive forces between all three particles in the system, and the repulsive force between the two protons is stronger than the attractive force between the electron and each proton.

I hope this helps to clarify the concept for you. Let me know if you have any further questions.